Calculus Examples

$8 (\frac{\tan^{3} (\frac{π}{6})}{3}) - 8 \frac{\tan^{3} (0)}{3}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$8 \frac{{(\frac{\sqrt{3}}{3})}^{3}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$8 \frac{\frac{{\sqrt{3}}^{3}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.3

Simplify the numerator.

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Step 1.1.3.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$8 \frac{\frac{\sqrt{3^{3}}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.3.2

Raise $3$ to the power of $3$ .

$8 \frac{\frac{\sqrt{27}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.3.3

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 1.1.3.3.1

Factor $9$ out of $27$ .

$8 \frac{\frac{\sqrt{9 (3)}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.3.3.2

Rewrite $9$ as $3^{2}$ .

$8 \frac{\frac{\sqrt{3^{2} \cdot 3}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.3.4

Pull terms out from under the radical.

$8 \frac{\frac{3 \sqrt{3}}{3^{3}}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.4

Raise $3$ to the power of $3$ .

$8 \frac{\frac{3 \sqrt{3}}{27}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.5

Cancel the common factor of $3$ and $27$ .

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Step 1.1.5.1

Factor $3$ out of $3 \sqrt{3}$ .

$8 \frac{\frac{3 (\sqrt{3})}{27}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.5.2

Cancel the common factors.

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Step 1.1.5.2.1

Factor $3$ out of $27$ .

$8 \frac{\frac{3 \sqrt{3}}{3 \cdot 9}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.5.2.2

Cancel the common factor.

$8 \frac{\frac{3 \sqrt{3}}{3 \cdot 9}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.1.5.2.3

Rewrite the expression.

$8 \frac{\frac{\sqrt{3}}{9}}{3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$8 (\frac{\sqrt{3}}{9} \cdot \frac{1}{3}) - 8 \frac{\tan^{3} (0)}{3}$

Step 1.3

Multiply $\frac{\sqrt{3}}{9} \cdot \frac{1}{3}$ .

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Step 1.3.1

Multiply $\frac{\sqrt{3}}{9}$ by $\frac{1}{3}$ .

$8 \frac{\sqrt{3}}{9 \cdot 3} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.3.2

Multiply $9$ by $3$ .

$8 \frac{\sqrt{3}}{27} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.4

Combine $8$ and $\frac{\sqrt{3}}{27}$ .

$\frac{8 \sqrt{3}}{27} - 8 \frac{\tan^{3} (0)}{3}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

The exact value of $\tan (0)$ is $0$ .

$\frac{8 \sqrt{3}}{27} - 8 \frac{0^{3}}{3}$

Step 1.5.2

Raising $0$ to any positive power yields $0$ .

$\frac{8 \sqrt{3}}{27} - 8 (\frac{0}{3})$

Step 1.6

Divide $0$ by $3$ .

$\frac{8 \sqrt{3}}{27} - 8 \cdot 0$

Step 1.7

Multiply $- 8$ by $0$ .

$\frac{8 \sqrt{3}}{27} + 0$

Step 2

Add $\frac{8 \sqrt{3}}{27}$ and $0$ .

$\frac{8 \sqrt{3}}{27}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{8 \sqrt{3}}{27}$

Decimal Form:

$0.51320023 \dots$